A reaction-diffusion population growth equation with multiple pulse perturbations

Abstract

During the pest growth generations there exist a number of internal and external perturbations including birth pulses and pulse pesticide applications, which result in a complex growth pattern within each pest growth generation due to the variation of perturbation timings. In order to show how multiple pulse perturbations affect the dynamics of the pest population, we have extended the classical Fisher reaction-diffusion equation by involving instant birth and control perturbations. The main results indicate that the pulse perturbations have a remarkable influence on the existence and stability of a spatially homogeneous periodic solution. In particular, the birth rate and killing efficacy can affect the traveling wave and its spreading speed, while the timing of pesticide application does not make any effects on those threshold conditions. However, numerical investigations reveal that the timing of pesticide applications can significantly affect the spatial distributions of the pest population. Within each generation too early or too late pesticide application could result in a faster growth and wider spread of the pest, and the optimal timing of pesticide application is the middle of each generation provided that the efficacy of the pesticide is relatively low. Bifurcation analyses reveal that even generations and odd generations display a different spatial pattern, which depict some important issues related to the pest control and pest management, for example the different control strategies should be adopted for even and odd generations. To address how the spatial heterogeneity affects on the cost of the pest control, we further formulate the cost function to investigate the optimal pest control strategy, and the optimal spatial dependent killing rates which minimize the cost have been obtained numerically.